Center for                           Nonlinear Analysis CNA Home People Seminars Publications Workshops and Conferences CNA Working Groups CNA Comments Form Summer Schools Summer Undergraduate Institute PIRE Cooperation Graduate Topics Courses SIAM Chapter Seminar Positions Contact Publication 16-CNA-010 A Second Order Minimality Condition for a Free-Boundary Problem Irene FonsecaDepartment of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213fonseca@andrew.cmu.edu Giovanni LeoniDepartment of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213giovanni@andrew.cmu.edu Maria Giovanna MoraDipartimento di Matematica Universita di Pavia Pavia, Italymariagiovanna.mora@unipv.itAbstract: The goal of this paper is to derive in the two-dimensional case necessary and sufficient minimality conditions in terms of the second variation for the functional $$v\mapsto\int_{\Omega}\big(|\nabla v|^{2}+\chi_{\{v>0\}}Q^{2}% \big)\,d\boldsymbol{x},$$ introduced in a classical paper of Alt and Caffarelli. For a special choice of $Q$ this includes water waves. The second variation is obtained by computing the second derivative of the functional along suitable variations of the free boundary. It is proved that the strict positivity of the second variation gives a sufficient condition for local minimality. Also, it is shown that smooth critical points are local minimizers in a small tubular neighborhood of the free-boundary.Get the paper in its entirety as  16-CNA-010.pdf